Introduction to Mathematical Analysis I - Second Edition

69 Theorem 3.2.2 (Cauchy’s criterion) Let f : D → R and let ¯ x be a limit point of D . Then f has a limit at ¯ x if and only if for any ε > 0, there exists δ > 0 such that | f ( r ) − f ( s ) | < ε whenever r , s ∈ D and 0 < | r − ¯ x | < δ , 0 < | s − ¯ x | < δ . (3.3) Proof: Suppose lim x → ¯ x f ( x ) = ` . Given ε > 0, there exists δ > 0 such that | f ( x ) − ` | < ε 2 whenever x ∈ D and 0 < | x − ¯ x | < δ . Thus, for r , s ∈ D with 0 < | r − ¯ x | < δ and 0 < | s − ¯ x | < δ , we have | f ( r ) − f ( s ) | ≤ | f ( r ) − ` | + | ` − f ( s ) | < ε . Let us prove the converse. Fix a sequence { u n } in D such with lim n → ∞ u n = ¯ x and u n 6 = ¯ x for every n . Given ε > 0, there exists δ > 0 such that | f ( r ) − f ( s ) | < ε whenever r , s ∈ D and 0 < | r − ¯ x | < δ , 0 < | s − ¯ x | < δ . Then there exists N ∈ N satisfying 0 < | u n − ¯ x | < δ for all n ≥ N . This implies | f ( u n ) − f ( u m ) | < ε for all m , n ≥ N . Thus, { f ( u n ) } is a Cauchy sequence, and hence there exists ` ∈ R such that lim n → ∞ f ( u n ) = `. We now prove that f has limit ` at ¯ x using Theorem 3.1.2 . Let { x n } be a sequence in D such that lim n → ∞ x n = ¯ x and x n 6 = ¯ x for every n . By the previous argument, there exists ` 0 ∈ R such that lim n → ∞ f ( x n ) = ` 0 . Fix any ε > 0 and let δ > 0 satisfy ( 3.3 ) . There exists K ∈ N such that | u n − ¯ x | < δ and | x n − ¯ x | < δ for all n ≥ K . Then | f ( u n ) − f ( x n ) | < ε for such n . Letting n → ∞ , we have | ` − ` 0 | ≤ ε . Thus, ` = ` 0 since ε is arbitrary. It now follows from Theorem 3.1.2 that lim x → ¯ x f ( x ) = ` . The rest of this section discussed some special limits and their properties. Definition 3.2.2 Let a ∈ R and δ > 0. Define B − ( a ; δ ) = ( a − δ , a ) and B + ( a ; δ ) = ( a , a + δ ) . Given a subset A of R , we say that a is a left limit point of A if for any δ > 0, B − ( a ; δ ) contains an infinite number of elements of A . Similarly, a is called a right limit point of A if for any δ > 0, B + ( a ; δ ) contains an infinite number of elements of A .